Overview
Factoring reverses multiplication and exposes a polynomial’s product structure. Start with a common factor; then look for a difference of squares or a trinomial whose factor coefficients reproduce both its middle and constant terms. A factorization is correct only when multiplying it out restores every coefficient.
Consider this example: Which product equals x² + 7x + 12? The constants 3 and 4 multiply to 12 and add to 7. Thus (x + 3)(x + 4) expands to the polynomial. The pair 2 and 6 has the right product but sum 8.
The method also handles a different case: Factor x² − 5x − 14. The needed constants multiply to −14 and add to −5: −7 and 2. Hence (x − 7)(x + 2) is correct. Reversing both signs changes the middle coefficient to +5.
For a further application, consider: Which product is a complete factorization of 2x³ + 6x² − 8x − 24? Group the terms: 2x²(x + 3) − 8(x + 3) = (x + 3)(2x² − 8). Extract 2 and factor the difference of squares, giving 2(x + 3)(x − 2)(x + 2). Expanding the first two groups checks the shared factor before the final split.
Finding numbers with the right product but the wrong sum gives a near-match rather than a factorization. Stopping after a common factor can leave a remaining factor that still splits. On a new polynomial, extract the greatest common factor first. Expand your proposed factors to check the middle term, which is often where the closest distractor differs.